src/HOL/Tools/cnf_funcs.ML
author wenzelm
Wed Mar 19 22:47:39 2008 +0100 (2008-03-19)
changeset 26341 2f5a4367a39e
parent 24958 ff15f76741bd
child 29265 5b4247055bd7
permissions -rw-r--r--
avoid Auto_tac;
webertj@17618
     1
(*  Title:      HOL/Tools/cnf_funcs.ML
webertj@17618
     2
    ID:         $Id$
webertj@17618
     3
    Author:     Alwen Tiu, QSL Team, LORIA (http://qsl.loria.fr)
webertj@17809
     4
    Author:     Tjark Weber
webertj@19236
     5
    Copyright   2005-2006
webertj@17618
     6
paulson@24958
     7
  FIXME: major overlaps with the code in meson.ML
paulson@24958
     8
webertj@17618
     9
  Description:
webertj@17809
    10
  This file contains functions and tactics to transform a formula into
webertj@17809
    11
  Conjunctive Normal Form (CNF).
webertj@17618
    12
  A formula in CNF is of the following form:
webertj@17618
    13
webertj@19236
    14
      (x11 | x12 | ... | x1n) & ... & (xm1 | xm2 | ... | xmk)
webertj@17809
    15
      False
webertj@17809
    16
      True
webertj@17618
    17
webertj@17809
    18
  where each xij is a literal (a positive or negative atomic Boolean term),
webertj@17809
    19
  i.e. the formula is a conjunction of disjunctions of literals, or
webertj@17809
    20
  "False", or "True".
webertj@17809
    21
webertj@17809
    22
  A (non-empty) disjunction of literals is referred to as "clause".
webertj@17618
    23
webertj@17618
    24
  For the purpose of SAT proof reconstruction, we also make use of another
webertj@17618
    25
  representation of clauses, which we call the "raw clauses".
webertj@17618
    26
  Raw clauses are of the form
webertj@17618
    27
webertj@20440
    28
      [..., x1', x2', ..., xn'] |- False ,
webertj@19236
    29
webertj@19236
    30
  where each xi is a literal, and each xi' is the negation normal form of ~xi.
webertj@17618
    31
webertj@19236
    32
  Literals are successively removed from the hyps of raw clauses by resolution
webertj@19236
    33
  during SAT proof reconstruction.
webertj@17618
    34
*)
webertj@17618
    35
webertj@17618
    36
signature CNF =
webertj@17618
    37
sig
webertj@17809
    38
	val is_atom           : Term.term -> bool
webertj@17809
    39
	val is_literal        : Term.term -> bool
webertj@17809
    40
	val is_clause         : Term.term -> bool
webertj@17809
    41
	val clause_is_trivial : Term.term -> bool
webertj@17809
    42
webertj@20440
    43
	val clause2raw_thm : Thm.thm -> Thm.thm
webertj@17618
    44
webertj@17809
    45
	val weakening_tac : int -> Tactical.tactic  (* removes the first hypothesis of a subgoal *)
webertj@17618
    46
wenzelm@20547
    47
	val make_cnf_thm  : theory -> Term.term -> Thm.thm
wenzelm@20547
    48
	val make_cnfx_thm : theory -> Term.term ->  Thm.thm
webertj@17809
    49
	val cnf_rewrite_tac  : int -> Tactical.tactic  (* converts all prems of a subgoal to CNF *)
webertj@17809
    50
	val cnfx_rewrite_tac : int -> Tactical.tactic  (* converts all prems of a subgoal to (almost) definitional CNF *)
webertj@17809
    51
end;
webertj@17618
    52
webertj@17618
    53
structure cnf : CNF =
webertj@17618
    54
struct
webertj@17618
    55
webertj@20440
    56
fun thm_by_auto (G : string) : thm =
wenzelm@26341
    57
	prove_goal (the_context ()) G (fn prems => [cut_facts_tac prems 1, CLASIMPSET auto_tac]);
webertj@17618
    58
webertj@17809
    59
(* Thm.thm *)
webertj@17809
    60
val clause2raw_notE      = thm_by_auto "[| P; ~P |] ==> False";
webertj@17809
    61
val clause2raw_not_disj  = thm_by_auto "[| ~P; ~Q |] ==> ~(P | Q)";
webertj@17809
    62
val clause2raw_not_not   = thm_by_auto "P ==> ~~P";
webertj@17618
    63
webertj@17809
    64
val iff_refl             = thm_by_auto "(P::bool) = P";
webertj@17809
    65
val iff_trans            = thm_by_auto "[| (P::bool) = Q; Q = R |] ==> P = R";
webertj@17809
    66
val conj_cong            = thm_by_auto "[| P = P'; Q = Q' |] ==> (P & Q) = (P' & Q')";
webertj@17809
    67
val disj_cong            = thm_by_auto "[| P = P'; Q = Q' |] ==> (P | Q) = (P' | Q')";
webertj@17618
    68
webertj@17809
    69
val make_nnf_imp         = thm_by_auto "[| (~P) = P'; Q = Q' |] ==> (P --> Q) = (P' | Q')";
webertj@17809
    70
val make_nnf_iff         = thm_by_auto "[| P = P'; (~P) = NP; Q = Q'; (~Q) = NQ |] ==> (P = Q) = ((P' | NQ) & (NP | Q'))";
webertj@17809
    71
val make_nnf_not_false   = thm_by_auto "(~False) = True";
webertj@17809
    72
val make_nnf_not_true    = thm_by_auto "(~True) = False";
webertj@17809
    73
val make_nnf_not_conj    = thm_by_auto "[| (~P) = P'; (~Q) = Q' |] ==> (~(P & Q)) = (P' | Q')";
webertj@17809
    74
val make_nnf_not_disj    = thm_by_auto "[| (~P) = P'; (~Q) = Q' |] ==> (~(P | Q)) = (P' & Q')";
webertj@17809
    75
val make_nnf_not_imp     = thm_by_auto "[| P = P'; (~Q) = Q' |] ==> (~(P --> Q)) = (P' & Q')";
webertj@17809
    76
val make_nnf_not_iff     = thm_by_auto "[| P = P'; (~P) = NP; Q = Q'; (~Q) = NQ |] ==> (~(P = Q)) = ((P' | Q') & (NP | NQ))";
webertj@17809
    77
val make_nnf_not_not     = thm_by_auto "P = P' ==> (~~P) = P'";
webertj@17618
    78
webertj@17809
    79
val simp_TF_conj_True_l  = thm_by_auto "[| P = True; Q = Q' |] ==> (P & Q) = Q'";
webertj@17809
    80
val simp_TF_conj_True_r  = thm_by_auto "[| P = P'; Q = True |] ==> (P & Q) = P'";
webertj@17809
    81
val simp_TF_conj_False_l = thm_by_auto "P = False ==> (P & Q) = False";
webertj@17809
    82
val simp_TF_conj_False_r = thm_by_auto "Q = False ==> (P & Q) = False";
webertj@17809
    83
val simp_TF_disj_True_l  = thm_by_auto "P = True ==> (P | Q) = True";
webertj@17809
    84
val simp_TF_disj_True_r  = thm_by_auto "Q = True ==> (P | Q) = True";
webertj@17809
    85
val simp_TF_disj_False_l = thm_by_auto "[| P = False; Q = Q' |] ==> (P | Q) = Q'";
webertj@17809
    86
val simp_TF_disj_False_r = thm_by_auto "[| P = P'; Q = False |] ==> (P | Q) = P'";
webertj@17618
    87
webertj@17809
    88
val make_cnf_disj_conj_l = thm_by_auto "[| (P | R) = PR; (Q | R) = QR |] ==> ((P & Q) | R) = (PR & QR)";
webertj@17809
    89
val make_cnf_disj_conj_r = thm_by_auto "[| (P | Q) = PQ; (P | R) = PR |] ==> (P | (Q & R)) = (PQ & PR)";
webertj@17618
    90
webertj@17809
    91
val make_cnfx_disj_ex_l = thm_by_auto "((EX (x::bool). P x) | Q) = (EX x. P x | Q)";
webertj@17809
    92
val make_cnfx_disj_ex_r = thm_by_auto "(P | (EX (x::bool). Q x)) = (EX x. P | Q x)";
webertj@17809
    93
val make_cnfx_newlit    = thm_by_auto "(P | Q) = (EX x. (P | x) & (Q | ~x))";
webertj@17809
    94
val make_cnfx_ex_cong   = thm_by_auto "(ALL (x::bool). P x = Q x) ==> (EX x. P x) = (EX x. Q x)";
webertj@17618
    95
webertj@17809
    96
val weakening_thm        = thm_by_auto "[| P; Q |] ==> Q";
webertj@17618
    97
webertj@17809
    98
val cnftac_eq_imp        = thm_by_auto "[| P = Q; P |] ==> Q";
webertj@17618
    99
webertj@17809
   100
(* Term.term -> bool *)
webertj@17809
   101
fun is_atom (Const ("False", _))                                           = false
webertj@17809
   102
  | is_atom (Const ("True", _))                                            = false
webertj@17809
   103
  | is_atom (Const ("op &", _) $ _ $ _)                                    = false
webertj@17809
   104
  | is_atom (Const ("op |", _) $ _ $ _)                                    = false
webertj@17809
   105
  | is_atom (Const ("op -->", _) $ _ $ _)                                  = false
webertj@17809
   106
  | is_atom (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ _ $ _) = false
webertj@17809
   107
  | is_atom (Const ("Not", _) $ _)                                         = false
webertj@17809
   108
  | is_atom _                                                              = true;
webertj@17618
   109
webertj@17809
   110
(* Term.term -> bool *)
webertj@17809
   111
fun is_literal (Const ("Not", _) $ x) = is_atom x
webertj@17809
   112
  | is_literal x                      = is_atom x;
webertj@17618
   113
webertj@17809
   114
(* Term.term -> bool *)
webertj@17809
   115
fun is_clause (Const ("op |", _) $ x $ y) = is_clause x andalso is_clause y
webertj@17809
   116
  | is_clause x                           = is_literal x;
webertj@17618
   117
webertj@17809
   118
(* ------------------------------------------------------------------------- *)
webertj@17809
   119
(* clause_is_trivial: a clause is trivially true if it contains both an atom *)
webertj@17809
   120
(*      and the atom's negation                                              *)
webertj@17809
   121
(* ------------------------------------------------------------------------- *)
webertj@17809
   122
webertj@17809
   123
(* Term.term -> bool *)
webertj@17618
   124
webertj@17809
   125
fun clause_is_trivial c =
webertj@17809
   126
	let
webertj@17809
   127
		(* Term.term -> Term.term *)
webertj@17809
   128
		fun dual (Const ("Not", _) $ x) = x
webertj@17809
   129
		  | dual x                      = HOLogic.Not $ x
webertj@17809
   130
		(* Term.term list -> bool *)
webertj@17809
   131
		fun has_duals []      = false
webertj@17809
   132
		  | has_duals (x::xs) = (dual x) mem xs orelse has_duals xs
webertj@17809
   133
	in
paulson@24958
   134
		has_duals (HOLogic.disjuncts c)
webertj@17809
   135
	end;
webertj@17618
   136
webertj@17809
   137
(* ------------------------------------------------------------------------- *)
webertj@17809
   138
(* clause2raw_thm: translates a clause into a raw clause, i.e.               *)
webertj@20440
   139
(*        [...] |- x1 | ... | xn                                             *)
webertj@17809
   140
(*      (where each xi is a literal) is translated to                        *)
webertj@20440
   141
(*        [..., x1', ..., xn'] |- False ,                                    *)
webertj@20440
   142
(*      where each xi' is the negation normal form of ~xi                    *)
webertj@17809
   143
(* ------------------------------------------------------------------------- *)
webertj@17618
   144
webertj@17809
   145
(* Thm.thm -> Thm.thm *)
webertj@17618
   146
webertj@20440
   147
fun clause2raw_thm clause =
webertj@17809
   148
let
webertj@17809
   149
	(* eliminates negated disjunctions from the i-th premise, possibly *)
webertj@17809
   150
	(* adding new premises, then continues with the (i+1)-th premise   *)
webertj@20440
   151
	(* int -> Thm.thm -> Thm.thm *)
webertj@20440
   152
	fun not_disj_to_prem i thm =
webertj@17809
   153
		if i > nprems_of thm then
webertj@17809
   154
			thm
webertj@17809
   155
		else
webertj@20440
   156
			not_disj_to_prem (i+1) (Seq.hd (REPEAT_DETERM (rtac clause2raw_not_disj i) thm))
webertj@20440
   157
	(* moves all premises to hyps, i.e. "[...] |- A1 ==> ... ==> An ==> B" *)
webertj@20440
   158
	(* becomes "[..., A1, ..., An] |- B"                                   *)
webertj@20440
   159
	(* Thm.thm -> Thm.thm *)
webertj@20440
   160
	fun prems_to_hyps thm =
webertj@20440
   161
		fold (fn cprem => fn thm' =>
webertj@20440
   162
			Thm.implies_elim thm' (Thm.assume cprem)) (cprems_of thm) thm
webertj@17809
   163
in
webertj@20440
   164
	(* [...] |- ~(x1 | ... | xn) ==> False *)
webertj@20440
   165
	(clause2raw_notE OF [clause])
webertj@20440
   166
	(* [...] |- ~x1 ==> ... ==> ~xn ==> False *)
webertj@20440
   167
	|> not_disj_to_prem 1
webertj@20440
   168
	(* [...] |- x1' ==> ... ==> xn' ==> False *)
webertj@20440
   169
	|> Seq.hd o TRYALL (rtac clause2raw_not_not)
webertj@20440
   170
	(* [..., x1', ..., xn'] |- False *)
webertj@20440
   171
	|> prems_to_hyps
webertj@17809
   172
end;
webertj@17618
   173
webertj@17809
   174
(* ------------------------------------------------------------------------- *)
webertj@17809
   175
(* inst_thm: instantiates a theorem with a list of terms                     *)
webertj@17809
   176
(* ------------------------------------------------------------------------- *)
webertj@17618
   177
webertj@17809
   178
fun inst_thm thy ts thm =
webertj@17809
   179
	instantiate' [] (map (SOME o cterm_of thy) ts) thm;
webertj@17618
   180
webertj@17809
   181
(* ------------------------------------------------------------------------- *)
webertj@17809
   182
(*                         Naive CNF transformation                          *)
webertj@17809
   183
(* ------------------------------------------------------------------------- *)
webertj@17618
   184
webertj@17809
   185
(* ------------------------------------------------------------------------- *)
webertj@17809
   186
(* make_nnf_thm: produces a theorem of the form t = t', where t' is the      *)
webertj@17809
   187
(*      negation normal form (i.e. negation only occurs in front of atoms)   *)
webertj@17809
   188
(*      of t; implications ("-->") and equivalences ("=" on bool) are        *)
webertj@17809
   189
(*      eliminated (possibly causing an exponential blowup)                  *)
webertj@17809
   190
(* ------------------------------------------------------------------------- *)
webertj@17809
   191
webertj@17809
   192
(* Theory.theory -> Term.term -> Thm.thm *)
webertj@17618
   193
webertj@17809
   194
fun make_nnf_thm thy (Const ("op &", _) $ x $ y) =
webertj@17809
   195
	let
webertj@17809
   196
		val thm1 = make_nnf_thm thy x
webertj@17809
   197
		val thm2 = make_nnf_thm thy y
webertj@17809
   198
	in
webertj@17809
   199
		conj_cong OF [thm1, thm2]
webertj@17809
   200
	end
webertj@17809
   201
  | make_nnf_thm thy (Const ("op |", _) $ x $ y) =
webertj@17809
   202
	let
webertj@17809
   203
		val thm1 = make_nnf_thm thy x
webertj@17809
   204
		val thm2 = make_nnf_thm thy y
webertj@17809
   205
	in
webertj@17809
   206
		disj_cong OF [thm1, thm2]
webertj@17809
   207
	end
webertj@17809
   208
  | make_nnf_thm thy (Const ("op -->", _) $ x $ y) =
webertj@17809
   209
	let
webertj@17809
   210
		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
webertj@17809
   211
		val thm2 = make_nnf_thm thy y
webertj@17809
   212
	in
webertj@17809
   213
		make_nnf_imp OF [thm1, thm2]
webertj@17809
   214
	end
webertj@17809
   215
  | make_nnf_thm thy (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ x $ y) =
webertj@17809
   216
	let
webertj@17809
   217
		val thm1 = make_nnf_thm thy x
webertj@17809
   218
		val thm2 = make_nnf_thm thy (HOLogic.Not $ x)
webertj@17809
   219
		val thm3 = make_nnf_thm thy y
webertj@17809
   220
		val thm4 = make_nnf_thm thy (HOLogic.Not $ y)
webertj@17809
   221
	in
webertj@17809
   222
		make_nnf_iff OF [thm1, thm2, thm3, thm4]
webertj@17809
   223
	end
webertj@17809
   224
  | make_nnf_thm thy (Const ("Not", _) $ Const ("False", _)) =
webertj@17809
   225
	make_nnf_not_false
webertj@17809
   226
  | make_nnf_thm thy (Const ("Not", _) $ Const ("True", _)) =
webertj@17809
   227
	make_nnf_not_true
webertj@17809
   228
  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op &", _) $ x $ y)) =
webertj@17809
   229
	let
webertj@17809
   230
		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
webertj@17809
   231
		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
webertj@17809
   232
	in
webertj@17809
   233
		make_nnf_not_conj OF [thm1, thm2]
webertj@17809
   234
	end
webertj@17809
   235
  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op |", _) $ x $ y)) =
webertj@17809
   236
	let
webertj@17809
   237
		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
webertj@17809
   238
		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
webertj@17809
   239
	in
webertj@17809
   240
		make_nnf_not_disj OF [thm1, thm2]
webertj@17809
   241
	end
webertj@17809
   242
  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op -->", _) $ x $ y)) =
webertj@17809
   243
	let
webertj@17809
   244
		val thm1 = make_nnf_thm thy x
webertj@17809
   245
		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
webertj@17809
   246
	in
webertj@17809
   247
		make_nnf_not_imp OF [thm1, thm2]
webertj@17809
   248
	end
webertj@17809
   249
  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ x $ y)) =
webertj@17809
   250
	let
webertj@17809
   251
		val thm1 = make_nnf_thm thy x
webertj@17809
   252
		val thm2 = make_nnf_thm thy (HOLogic.Not $ x)
webertj@17809
   253
		val thm3 = make_nnf_thm thy y
webertj@17809
   254
		val thm4 = make_nnf_thm thy (HOLogic.Not $ y)
webertj@17809
   255
	in
webertj@17809
   256
		make_nnf_not_iff OF [thm1, thm2, thm3, thm4]
webertj@17809
   257
	end
webertj@17809
   258
  | make_nnf_thm thy (Const ("Not", _) $ (Const ("Not", _) $ x)) =
webertj@17809
   259
	let
webertj@17809
   260
		val thm1 = make_nnf_thm thy x
webertj@17809
   261
	in
webertj@17809
   262
		make_nnf_not_not OF [thm1]
webertj@17809
   263
	end
webertj@17809
   264
  | make_nnf_thm thy t =
webertj@17809
   265
	inst_thm thy [t] iff_refl;
webertj@17618
   266
webertj@17809
   267
(* ------------------------------------------------------------------------- *)
webertj@17809
   268
(* simp_True_False_thm: produces a theorem t = t', where t' is equivalent to *)
webertj@17809
   269
(*      t, but simplified wrt. the following theorems:                       *)
webertj@17809
   270
(*        (True & x) = x                                                     *)
webertj@17809
   271
(*        (x & True) = x                                                     *)
webertj@17809
   272
(*        (False & x) = False                                                *)
webertj@17809
   273
(*        (x & False) = False                                                *)
webertj@17809
   274
(*        (True | x) = True                                                  *)
webertj@17809
   275
(*        (x | True) = True                                                  *)
webertj@17809
   276
(*        (False | x) = x                                                    *)
webertj@17809
   277
(*        (x | False) = x                                                    *)
webertj@17809
   278
(*      No simplification is performed below connectives other than & and |. *)
webertj@17809
   279
(*      Optimization: The right-hand side of a conjunction (disjunction) is  *)
webertj@17809
   280
(*      simplified only if the left-hand side does not simplify to False     *)
webertj@17809
   281
(*      (True, respectively).                                                *)
webertj@17809
   282
(* ------------------------------------------------------------------------- *)
webertj@17618
   283
webertj@17809
   284
(* Theory.theory -> Term.term -> Thm.thm *)
webertj@17618
   285
webertj@17809
   286
fun simp_True_False_thm thy (Const ("op &", _) $ x $ y) =
webertj@17809
   287
	let
webertj@17809
   288
		val thm1 = simp_True_False_thm thy x
webertj@17809
   289
		val x'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
webertj@17809
   290
	in
webertj@17809
   291
		if x' = HOLogic.false_const then
webertj@17809
   292
			simp_TF_conj_False_l OF [thm1]  (* (x & y) = False *)
webertj@17809
   293
		else
webertj@17809
   294
			let
webertj@17809
   295
				val thm2 = simp_True_False_thm thy y
webertj@17809
   296
				val y'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
webertj@17809
   297
			in
webertj@17809
   298
				if x' = HOLogic.true_const then
webertj@17809
   299
					simp_TF_conj_True_l OF [thm1, thm2]  (* (x & y) = y' *)
webertj@17809
   300
				else if y' = HOLogic.false_const then
webertj@17809
   301
					simp_TF_conj_False_r OF [thm2]  (* (x & y) = False *)
webertj@17809
   302
				else if y' = HOLogic.true_const then
webertj@17809
   303
					simp_TF_conj_True_r OF [thm1, thm2]  (* (x & y) = x' *)
webertj@17809
   304
				else
webertj@17809
   305
					conj_cong OF [thm1, thm2]  (* (x & y) = (x' & y') *)
webertj@17809
   306
			end
webertj@17809
   307
	end
webertj@17809
   308
  | simp_True_False_thm thy (Const ("op |", _) $ x $ y) =
webertj@17809
   309
	let
webertj@17809
   310
		val thm1 = simp_True_False_thm thy x
webertj@17809
   311
		val x'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
webertj@17809
   312
	in
webertj@17809
   313
		if x' = HOLogic.true_const then
webertj@17809
   314
			simp_TF_disj_True_l OF [thm1]  (* (x | y) = True *)
webertj@17809
   315
		else
webertj@17809
   316
			let
webertj@17809
   317
				val thm2 = simp_True_False_thm thy y
webertj@17809
   318
				val y'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
webertj@17809
   319
			in
webertj@17809
   320
				if x' = HOLogic.false_const then
webertj@17809
   321
					simp_TF_disj_False_l OF [thm1, thm2]  (* (x | y) = y' *)
webertj@17809
   322
				else if y' = HOLogic.true_const then
webertj@17809
   323
					simp_TF_disj_True_r OF [thm2]  (* (x | y) = True *)
webertj@17809
   324
				else if y' = HOLogic.false_const then
webertj@17809
   325
					simp_TF_disj_False_r OF [thm1, thm2]  (* (x | y) = x' *)
webertj@17809
   326
				else
webertj@17809
   327
					disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
webertj@17809
   328
			end
webertj@17809
   329
	end
webertj@17809
   330
  | simp_True_False_thm thy t =
webertj@17809
   331
	inst_thm thy [t] iff_refl;  (* t = t *)
webertj@17618
   332
webertj@17809
   333
(* ------------------------------------------------------------------------- *)
webertj@17809
   334
(* make_cnf_thm: given any HOL term 't', produces a theorem t = t', where t' *)
webertj@17809
   335
(*      is in conjunction normal form.  May cause an exponential blowup      *)
webertj@17809
   336
(*      in the length of the term.                                           *)
webertj@17809
   337
(* ------------------------------------------------------------------------- *)
webertj@17618
   338
webertj@17809
   339
(* Theory.theory -> Term.term -> Thm.thm *)
webertj@17618
   340
webertj@17809
   341
fun make_cnf_thm thy t =
webertj@17809
   342
let
webertj@17809
   343
	(* Term.term -> Thm.thm *)
webertj@17809
   344
	fun make_cnf_thm_from_nnf (Const ("op &", _) $ x $ y) =
webertj@17809
   345
		let
webertj@17809
   346
			val thm1 = make_cnf_thm_from_nnf x
webertj@17809
   347
			val thm2 = make_cnf_thm_from_nnf y
webertj@17809
   348
		in
webertj@17809
   349
			conj_cong OF [thm1, thm2]
webertj@17809
   350
		end
webertj@17809
   351
	  | make_cnf_thm_from_nnf (Const ("op |", _) $ x $ y) =
webertj@17809
   352
		let
webertj@17809
   353
			(* produces a theorem "(x' | y') = t'", where x', y', and t' are in CNF *)
webertj@17809
   354
			fun make_cnf_disj_thm (Const ("op &", _) $ x1 $ x2) y' =
webertj@17809
   355
				let
webertj@17809
   356
					val thm1 = make_cnf_disj_thm x1 y'
webertj@17809
   357
					val thm2 = make_cnf_disj_thm x2 y'
webertj@17809
   358
				in
webertj@17809
   359
					make_cnf_disj_conj_l OF [thm1, thm2]  (* ((x1 & x2) | y') = ((x1 | y')' & (x2 | y')') *)
webertj@17809
   360
				end
webertj@17809
   361
			  | make_cnf_disj_thm x' (Const ("op &", _) $ y1 $ y2) =
webertj@17809
   362
				let
webertj@17809
   363
					val thm1 = make_cnf_disj_thm x' y1
webertj@17809
   364
					val thm2 = make_cnf_disj_thm x' y2
webertj@17809
   365
				in
webertj@17809
   366
					make_cnf_disj_conj_r OF [thm1, thm2]  (* (x' | (y1 & y2)) = ((x' | y1)' & (x' | y2)') *)
webertj@17809
   367
				end
webertj@17809
   368
			  | make_cnf_disj_thm x' y' =
webertj@17809
   369
				inst_thm thy [HOLogic.mk_disj (x', y')] iff_refl  (* (x' | y') = (x' | y') *)
webertj@17809
   370
			val thm1     = make_cnf_thm_from_nnf x
webertj@17809
   371
			val thm2     = make_cnf_thm_from_nnf y
webertj@17809
   372
			val x'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
webertj@17809
   373
			val y'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
webertj@17809
   374
			val disj_thm = disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
webertj@17809
   375
		in
webertj@17809
   376
			iff_trans OF [disj_thm, make_cnf_disj_thm x' y']
webertj@17809
   377
		end
webertj@17809
   378
	  | make_cnf_thm_from_nnf t =
webertj@17809
   379
		inst_thm thy [t] iff_refl
webertj@17809
   380
	(* convert 't' to NNF first *)
webertj@17809
   381
	val nnf_thm  = make_nnf_thm thy t
webertj@17809
   382
	val nnf      = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) nnf_thm
webertj@17809
   383
	(* then simplify wrt. True/False (this should preserve NNF) *)
webertj@17809
   384
	val simp_thm = simp_True_False_thm thy nnf
webertj@17809
   385
	val simp     = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) simp_thm
webertj@17809
   386
	(* finally, convert to CNF (this should preserve the simplification) *)
webertj@17809
   387
	val cnf_thm  = make_cnf_thm_from_nnf simp
webertj@17618
   388
in
webertj@17809
   389
	iff_trans OF [iff_trans OF [nnf_thm, simp_thm], cnf_thm]
webertj@17809
   390
end;
webertj@17618
   391
webertj@17809
   392
(* ------------------------------------------------------------------------- *)
webertj@17809
   393
(*            CNF transformation by introducing new literals                 *)
webertj@17809
   394
(* ------------------------------------------------------------------------- *)
webertj@17618
   395
webertj@17809
   396
(* ------------------------------------------------------------------------- *)
webertj@17809
   397
(* make_cnfx_thm: given any HOL term 't', produces a theorem t = t', where   *)
webertj@17809
   398
(*      t' is almost in conjunction normal form, except that conjunctions    *)
webertj@17809
   399
(*      and existential quantifiers may be nested.  (Use e.g. 'REPEAT_DETERM *)
webertj@17809
   400
(*      (etac exE i ORELSE etac conjE i)' afterwards to normalize.)  May     *)
webertj@17809
   401
(*      introduce new (existentially bound) literals.  Note: the current     *)
webertj@17809
   402
(*      implementation calls 'make_nnf_thm', causing an exponential blowup   *)
webertj@17809
   403
(*      in the case of nested equivalences.                                  *)
webertj@17809
   404
(* ------------------------------------------------------------------------- *)
webertj@17618
   405
webertj@17809
   406
(* Theory.theory -> Term.term -> Thm.thm *)
webertj@17618
   407
webertj@17809
   408
fun make_cnfx_thm thy t =
webertj@17809
   409
let
webertj@17809
   410
	val var_id = ref 0  (* properly initialized below *)
webertj@17809
   411
	(* unit -> Term.term *)
webertj@17809
   412
	fun new_free () =
webertj@17809
   413
		Free ("cnfx_" ^ string_of_int (inc var_id), HOLogic.boolT)
webertj@17809
   414
	(* Term.term -> Thm.thm *)
webertj@17809
   415
	fun make_cnfx_thm_from_nnf (Const ("op &", _) $ x $ y) =
webertj@17809
   416
		let
webertj@17809
   417
			val thm1 = make_cnfx_thm_from_nnf x
webertj@17809
   418
			val thm2 = make_cnfx_thm_from_nnf y
webertj@17809
   419
		in
webertj@17809
   420
			conj_cong OF [thm1, thm2]
webertj@17809
   421
		end
webertj@17809
   422
	  | make_cnfx_thm_from_nnf (Const ("op |", _) $ x $ y) =
webertj@17809
   423
		if is_clause x andalso is_clause y then
webertj@17809
   424
			inst_thm thy [HOLogic.mk_disj (x, y)] iff_refl
webertj@17809
   425
		else if is_literal y orelse is_literal x then let
webertj@17809
   426
			(* produces a theorem "(x' | y') = t'", where x', y', and t' are *)
webertj@17809
   427
			(* almost in CNF, and x' or y' is a literal                      *)
webertj@17809
   428
			fun make_cnfx_disj_thm (Const ("op &", _) $ x1 $ x2) y' =
webertj@17809
   429
				let
webertj@17809
   430
					val thm1 = make_cnfx_disj_thm x1 y'
webertj@17809
   431
					val thm2 = make_cnfx_disj_thm x2 y'
webertj@17809
   432
				in
webertj@17809
   433
					make_cnf_disj_conj_l OF [thm1, thm2]  (* ((x1 & x2) | y') = ((x1 | y')' & (x2 | y')') *)
webertj@17809
   434
				end
webertj@17809
   435
			  | make_cnfx_disj_thm x' (Const ("op &", _) $ y1 $ y2) =
webertj@17809
   436
				let
webertj@17809
   437
					val thm1 = make_cnfx_disj_thm x' y1
webertj@17809
   438
					val thm2 = make_cnfx_disj_thm x' y2
webertj@17809
   439
				in
webertj@17809
   440
					make_cnf_disj_conj_r OF [thm1, thm2]  (* (x' | (y1 & y2)) = ((x' | y1)' & (x' | y2)') *)
webertj@17809
   441
				end
webertj@17809
   442
			  | make_cnfx_disj_thm (Const ("Ex", _) $ x') y' =
webertj@17809
   443
				let
webertj@17809
   444
					val thm1 = inst_thm thy [x', y'] make_cnfx_disj_ex_l   (* ((Ex x') | y') = (Ex (x' | y')) *)
webertj@17809
   445
					val var  = new_free ()
webertj@17809
   446
					val thm2 = make_cnfx_disj_thm (betapply (x', var)) y'  (* (x' | y') = body' *)
webertj@17809
   447
					val thm3 = forall_intr (cterm_of thy var) thm2         (* !!v. (x' | y') = body' *)
webertj@17809
   448
					val thm4 = strip_shyps (thm3 COMP allI)                (* ALL v. (x' | y') = body' *)
webertj@17809
   449
					val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)     (* (EX v. (x' | y')) = (EX v. body') *)
webertj@17809
   450
				in
webertj@17809
   451
					iff_trans OF [thm1, thm5]  (* ((Ex x') | y') = (Ex v. body') *)
webertj@17809
   452
				end
webertj@17809
   453
			  | make_cnfx_disj_thm x' (Const ("Ex", _) $ y') =
webertj@17809
   454
				let
webertj@17809
   455
					val thm1 = inst_thm thy [x', y'] make_cnfx_disj_ex_r   (* (x' | (Ex y')) = (Ex (x' | y')) *)
webertj@17809
   456
					val var  = new_free ()
webertj@17809
   457
					val thm2 = make_cnfx_disj_thm x' (betapply (y', var))  (* (x' | y') = body' *)
webertj@17809
   458
					val thm3 = forall_intr (cterm_of thy var) thm2         (* !!v. (x' | y') = body' *)
webertj@17809
   459
					val thm4 = strip_shyps (thm3 COMP allI)                (* ALL v. (x' | y') = body' *)
webertj@17809
   460
					val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)     (* (EX v. (x' | y')) = (EX v. body') *)
webertj@17809
   461
				in
webertj@17809
   462
					iff_trans OF [thm1, thm5]  (* (x' | (Ex y')) = (EX v. body') *)
webertj@17809
   463
				end
webertj@17809
   464
			  | make_cnfx_disj_thm x' y' =
webertj@17809
   465
				inst_thm thy [HOLogic.mk_disj (x', y')] iff_refl  (* (x' | y') = (x' | y') *)
webertj@17809
   466
			val thm1     = make_cnfx_thm_from_nnf x
webertj@17809
   467
			val thm2     = make_cnfx_thm_from_nnf y
webertj@17809
   468
			val x'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
webertj@17809
   469
			val y'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
webertj@17809
   470
			val disj_thm = disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
webertj@17809
   471
		in
webertj@17809
   472
			iff_trans OF [disj_thm, make_cnfx_disj_thm x' y']
webertj@17809
   473
		end else let  (* neither 'x' nor 'y' is a literal: introduce a fresh variable *)
webertj@17809
   474
			val thm1 = inst_thm thy [x, y] make_cnfx_newlit     (* (x | y) = EX v. (x | v) & (y | ~v) *)
webertj@17809
   475
			val var  = new_free ()
webertj@17809
   476
			val body = HOLogic.mk_conj (HOLogic.mk_disj (x, var), HOLogic.mk_disj (y, HOLogic.Not $ var))
webertj@17809
   477
			val thm2 = make_cnfx_thm_from_nnf body              (* (x | v) & (y | ~v) = body' *)
webertj@17809
   478
			val thm3 = forall_intr (cterm_of thy var) thm2      (* !!v. (x | v) & (y | ~v) = body' *)
webertj@17809
   479
			val thm4 = strip_shyps (thm3 COMP allI)             (* ALL v. (x | v) & (y | ~v) = body' *)
webertj@17809
   480
			val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)  (* (EX v. (x | v) & (y | ~v)) = (EX v. body') *)
webertj@17809
   481
		in
webertj@17809
   482
			iff_trans OF [thm1, thm5]
webertj@17809
   483
		end
webertj@17809
   484
	  | make_cnfx_thm_from_nnf t =
webertj@17809
   485
		inst_thm thy [t] iff_refl
webertj@17809
   486
	(* convert 't' to NNF first *)
webertj@17809
   487
	val nnf_thm  = make_nnf_thm thy t
webertj@17809
   488
	val nnf      = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) nnf_thm
webertj@17809
   489
	(* then simplify wrt. True/False (this should preserve NNF) *)
webertj@17809
   490
	val simp_thm = simp_True_False_thm thy nnf
webertj@17809
   491
	val simp     = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) simp_thm
webertj@17809
   492
	(* initialize var_id, in case the term already contains variables of the form "cnfx_<int>" *)
webertj@17809
   493
	val _        = (var_id := fold (fn free => fn max =>
webertj@17809
   494
		let
webertj@17809
   495
			val (name, _) = dest_Free free
webertj@17809
   496
			val idx       = if String.isPrefix "cnfx_" name then
webertj@17809
   497
					(Int.fromString o String.extract) (name, String.size "cnfx_", NONE)
webertj@17809
   498
				else
webertj@17809
   499
					NONE
webertj@17809
   500
		in
webertj@17809
   501
			Int.max (max, getOpt (idx, 0))
webertj@17809
   502
		end) (term_frees simp) 0)
webertj@17809
   503
	(* finally, convert to definitional CNF (this should preserve the simplification) *)
webertj@17809
   504
	val cnfx_thm = make_cnfx_thm_from_nnf simp
webertj@17809
   505
in
webertj@17809
   506
	iff_trans OF [iff_trans OF [nnf_thm, simp_thm], cnfx_thm]
webertj@17809
   507
end;
webertj@17618
   508
webertj@17809
   509
(* ------------------------------------------------------------------------- *)
webertj@17809
   510
(*                                  Tactics                                  *)
webertj@17809
   511
(* ------------------------------------------------------------------------- *)
webertj@17618
   512
webertj@17809
   513
(* ------------------------------------------------------------------------- *)
webertj@17809
   514
(* weakening_tac: removes the first hypothesis of the 'i'-th subgoal         *)
webertj@17809
   515
(* ------------------------------------------------------------------------- *)
webertj@17618
   516
webertj@17809
   517
(* int -> Tactical.tactic *)
webertj@17618
   518
webertj@17809
   519
fun weakening_tac i =
webertj@17809
   520
	dtac weakening_thm i THEN atac (i+1);
webertj@17618
   521
webertj@17809
   522
(* ------------------------------------------------------------------------- *)
webertj@17809
   523
(* cnf_rewrite_tac: converts all premises of the 'i'-th subgoal to CNF       *)
webertj@17809
   524
(*      (possibly causing an exponential blowup in the length of each        *)
webertj@17809
   525
(*      premise)                                                             *)
webertj@17809
   526
(* ------------------------------------------------------------------------- *)
webertj@17618
   527
webertj@17809
   528
(* int -> Tactical.tactic *)
webertj@17618
   529
webertj@17809
   530
fun cnf_rewrite_tac i =
webertj@17809
   531
	(* cut the CNF formulas as new premises *)
webertj@17809
   532
	METAHYPS (fn prems =>
webertj@17809
   533
		let
webertj@17809
   534
			val cnf_thms = map (fn pr => make_cnf_thm (theory_of_thm pr) ((HOLogic.dest_Trueprop o prop_of) pr)) prems
webertj@17809
   535
			val cut_thms = map (fn (th, pr) => cnftac_eq_imp OF [th, pr]) (cnf_thms ~~ prems)
webertj@17809
   536
		in
webertj@17809
   537
			cut_facts_tac cut_thms 1
webertj@17809
   538
		end) i
webertj@17809
   539
	(* remove the original premises *)
webertj@17809
   540
	THEN SELECT_GOAL (fn thm =>
webertj@17809
   541
		let
wenzelm@21576
   542
			val n = Logic.count_prems ((Term.strip_all_body o fst o Logic.dest_implies o prop_of) thm)
webertj@17809
   543
		in
webertj@17809
   544
			PRIMITIVE (funpow (n div 2) (Seq.hd o weakening_tac 1)) thm
webertj@17809
   545
		end) i;
webertj@17618
   546
webertj@17809
   547
(* ------------------------------------------------------------------------- *)
webertj@17809
   548
(* cnfx_rewrite_tac: converts all premises of the 'i'-th subgoal to CNF      *)
webertj@17809
   549
(*      (possibly introducing new literals)                                  *)
webertj@17809
   550
(* ------------------------------------------------------------------------- *)
webertj@17809
   551
webertj@17809
   552
(* int -> Tactical.tactic *)
webertj@17618
   553
webertj@17809
   554
fun cnfx_rewrite_tac i =
webertj@17809
   555
	(* cut the CNF formulas as new premises *)
webertj@17809
   556
	METAHYPS (fn prems =>
webertj@17809
   557
		let
webertj@17809
   558
			val cnfx_thms = map (fn pr => make_cnfx_thm (theory_of_thm pr) ((HOLogic.dest_Trueprop o prop_of) pr)) prems
webertj@17809
   559
			val cut_thms  = map (fn (th, pr) => cnftac_eq_imp OF [th, pr]) (cnfx_thms ~~ prems)
webertj@17809
   560
		in
webertj@17809
   561
			cut_facts_tac cut_thms 1
webertj@17809
   562
		end) i
webertj@17809
   563
	(* remove the original premises *)
webertj@17809
   564
	THEN SELECT_GOAL (fn thm =>
webertj@17809
   565
		let
wenzelm@21576
   566
			val n = Logic.count_prems ((Term.strip_all_body o fst o Logic.dest_implies o prop_of) thm)
webertj@17809
   567
		in
webertj@17809
   568
			PRIMITIVE (funpow (n div 2) (Seq.hd o weakening_tac 1)) thm
webertj@17809
   569
		end) i;
webertj@17618
   570
webertj@17809
   571
end;  (* of structure *)